We describe an algorithmic reduction of the search for integral points
on a curve y2 =
ax4 + bx2 + c
with ac(b2 - 4ac) ≠ 0 to solving a
finite number of Thue equations. While the existence of such a reduction is
anticipated from arguments of algebraic number theory, our algorithm is
elementary and is, to the best of our knowledge, the first published
algorithm of this kind. In combination with other methods and powered
by existing Thue equation solvers, it allows one to
efficiently compute integral points on biquadratic curves.

We illustrate this approach with a particular application of finding near-multiples of squares in Lucas sequences.
As an example, we establish that among Fibonacci numbers only 2 and 34
are of the form 2m2+2; only 1, 13, and 1597 are of
the form m2-3; and so on.

As an auxiliary result, we also give an algorithm for solving a
Diophantine equation k^2 =
f(m,n)/g(m,n) in
integers m, n, k, where f and g are
homogeneous quadratic polynomials.